# Estimation of market price of risk of short interest rate under the Hull-White model

I think I am a bit confused. I intend to estimate the market price of risk the short interest rate, say, under the Hull-White model. I have the following two questions.

1. Is it correct to state state that the (zero coupon) bond price and the (zero coupon) bond European option price are both valued in the risk-neutral measure, because they are both derivatives of the short rate? If so, the parameters calibrated from these prices would not provide the market price of risk since the latter could only be estimated in the real world measure. Is this correct?

2. If the answer to the above question is affirmative, how does one estimate the market price of risk (risk premium) of the short interest rate?

There's a large literature dedicated to this topic. The following two methods are my preferred (they also happen to be the most popular):

1. Kim and Orphanides's Term Structure Estimation with Survey Data on Interest Rate Forecasts or Kim and Wright's An Arbitrage-Free Three-Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates. The method described in these two papers are pretty generic – the authors calibrate model parameters under both the physical and risk neutral measure by means of Kalman filtering. Difference in interest rates calculated under the two measures then reflect the market price of risk. The innovation of these papers is the introduction of survey data to anchor down the expectation component, so the results are more robust and realistic. Both papers are based on generalized affine term structure models, of which Hull White is a special case.
2. A more recent method is proposed by Adrian, Crump, and Moench in Pricing the Term Structure with Linear Regressions. Instead of Kalman filtering, they use simple regression techniques to obtain model parameters in the two measures. Instead of using survey data to pin down rate expectations, they use realized bond excess returns to pin down ex-ante risk premium.
• Thank you for answering my question #2. Could you please have a go at my question #1?
– Hans
Aug 16 '18 at 7:11
• @Hans That's correct. Quoted market prices from a single trading day is insufficient for inferring parameters from real world measures. Aug 16 '18 at 7:41
• +1. Thank you. Regarding your answer #2, suppose the short rate follows the Hull-White model, what determines the realized bond excess return and how is it related to the short rate risk premium? The Hull-White model implies the bond price $P(t,T)$ follows $\frac{dP(t,T)}{P(t,T)}=r(t)\,dt−\sigma_0 B(t,T)\,dW_t$. So the return is just the short interest rate $r$. Is there an extra term to $r$ to result in the excess return?
– Hans
Aug 16 '18 at 17:51
• The market price of risk reflects ex-ante / expected risk premium (i.e., excess return). Realized excess return is calculated directly from historical data (ACM uses zero coupon bonds; they detail their methodology in the paper). The objective is to calibrate model parameters such that the ex-ante risk premium dynamic mirror realized excess returns as closely as the model allows. You can find the full derivation for expected excess returns in these papers as well. Aug 17 '18 at 18:32
• Here is the correction to the confusing remark in my last comment. $dP=rPdt+\sigma\frac{\partial P}{\partial r}dW_Q=\Big(rP+\lambda\sigma\frac{\partial P}{\partial r}\Big)dt+\sigma\frac{\partial P}{\partial r}dW$, where $W$ and $W_Q$ stand for the Brownian motions in the physical measure and the risk neutral measure $Q$, respectively, and $\sigma$ for the instantaneous volatility of the short rate $r$, $\lambda$ the interest rate risk premium. So $\lambda\sigma\frac1P\frac{\partial P}{\partial r}$ is the excess return in the physical measure.
– Hans
Aug 26 '18 at 3:35